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Microsoft Word - ELEC321 Assignment 1 2018 Semester 2 Page 1 Department of Engineering ELEC321/621 Communications Systems, Semester 2, 2018 Assignment 1 Completion of Assignment 1 is worth 10% of your total marks for this unit. Your Assignment 1 must be submitted on the ELEC321 iLearn website by 11:55 p.m. on Sunday, 9 September 2018. The submission should be made using the Assignments drop box feature in iLearn. The Assignments button can be found on the left‐hand side of the webpage after logging into the ELEC321 Electronic ilearn unit. The report should be submitted in the PDF format (scanned hand‐written solutions are also acceptable but the hand writing must be legible). The following file naming format should be used: FamilyName_GivenName_Assignment1.pdf Late submissions will incur a penalty of 10% for each working day or part of beyond the submission deadline. A late submission of more than 7 days will receive a mark of 0. A completed and signed cover sheet available on the unit website must be submitted with the report. A scanned version of the completed cover sheet may be attached in front of the report (forming a single pdf document). Plagiarism will result in 0 marks in the assignment for all parties concerned and can lead to failure in the unit and reference to the University Discipline Committee. For the University’s honesty policy see www.student.mq.edu.au/plagiarism Page 2 Question 1 (20 Marks=2.5 +2.5 +10+2.5+2.5) is a periodic function which is given by,x t , 12 4 , - 0, 2 1where, 2 4 , 0 , 2 0, elsewhere. k x t g t k t t g t t t A. What is the period of the periodic function, ?x t B. Use Matlab to plot as a function of for -6 6.x t t t C. is the complex Fourier Series representation of FSx t x t ? Find a mathematical expression for FSx t . D. Use Matlab to plot and for ‐1 1 on the same graph.FSx t x t t Use Fourier Series Coefficients for -101 k 101 for the evaluation of Fourier Series. E. Parseval’s Theorem States that, 2 22 2 2 2 1 , where is the period of , is the average power of , and the complex Fourier Series representation of given by ˆUse Matlab to plot T k av T av ktj T k m av k k m x t dt c P T T x t P x t x t x t c e P m c as a function of for 1 12. ˆOn the same plot, plot to compare to .av av m m P P m Page 3 Question 2 (20 Marks =7+6+7) 2 is a modulated real bandpass signal. is a message signal. can be expressed by an complex envelope representation given by, Re , where is the complex envelope of c BP BP j f t BP BP x t m t x t x t x t e x t x 2 and is the centre frequency of . Re is the real part. A. 1 cos 2 . What is the complex envelope, , such that Re . B. = Re 2 c c BP BP AM c j f t BP AM BP USB t f x x t A m t f t x t x t x t e Ax t m 2 2 0 ˆ ˆ , where is the Hilbert transform of . What is the complex envelope, , such that Re . C. cos 2 2 . c c j f t j f t BP USB t BP FM c o t jm t e m t m t x t x t x t e x t A f t K m t dt 2 What is the complex envelope, , such that Re cj f tBP FMx t x t x t e Page 4 Question 3 (20 Marks=5+5+5+5) The function in Matlab is useful to plotting functions which are defined at discrete points. stem •HINT : is defined by, 10 cos 2 400 t +5 cos 2 1000 t +2 cos 2 3000 A. Plot the amplitude spectrum as a function of for the message signal . For parts B, C, and D it is only required to plot m t m t t M f f m t Hint : 5 the amplitude spectrum for 0. B. is the input to a DSB-FC modulator which is characterized by, 1 ( ) 20 1+ cos 2 10 t 17 Use Matlab to plot the amplitude spectrum, , w DSB FC DSB FC f m t s t m t S f 5 here ( ) and is the Fourier Transform. C. is the input to a DSB-SC modulator which is characterized by, 20 ( ) cos 2 10 t 17 Use Matlab to plot the am DSB FC DSB FC DSB SC S f s t m t s t m t F F 5 plitude spectrum, , where ( ) and is the Fourier Transform. D. is the input to a SSB-USB modulator which is characterized by, 10 ˆ ( ) cos 2 10 t 17 DSB SC DSB SC DSB SCS f S f s t m t s t m t jm t F F ˆ, where is the Hilbert Transform of . Use Matlab to plot the amplitude spectrum, , where ( ) and is the Fourier Transform. m t m t S f S f s tF F Page 5 Question 4 (20 Marks=5+5+5+5) A modulated bandpass signal can be characterized in terms of its complex envelope. Therefore, 2 cos 2 f t + Re e , where is the complex envelope of ( ), is the envelope of , cf t cs t A t t s t s t s t A t s t 1 and is the phase angle of . A. Prove that Im B. Prove that tan Re C. If is received at a cha t s t E t A t s t s t t s t s t nnel output, indicate the signal processing required to obtain at the receiver. Assume that cos 2 f t and sin 2 f t are available at the receiver. D. If , cos 2 f t , and sin 2 f t are availa c c c c s t s t ble at the modulator, indicate the signal processing required to obtain .s t Page 6 Question 5 (20 Marks=5+3+5+3+2+2) 2 2 The autocorrelation function of a periodic signal with period is given by 1 The power spectral density of is given by , where is the Fourier Transform. A. For T x T X X x t T R x t x t dt T x t S f R F F 9 sin 2 , find . B. Use Matlab to plot . Plot the case of 1 volt, 1. 10 Hz, and - , 2 2 1 where . C. For sin 2 , find . D. Use Matlab to plot . Pl c x x c c c x x x t A f t R t T TR A f T f x t A f t S f S f 9 9 9 2 2 2 ot the case of 1 volt, 1. 10 Hz, and -2 10 Hz 2 10 Hz. 1E. What is the relationship between and the signal power, ? F. What is the relationship between an c T x x T X A f f S f P x t dt T R d ?xP